Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

SECTION 13.2 DERIVATIVES AND INTEGRALS OF VECTOR FUNCTIONS 0 157
13.2 Derivatives and Integrals of Vector Functions
1. (a)
r(4.5) - r(4)
Q
r(4.2)- r(4)
)(.
(b) r( 4·5)- r ( 4 ) = 2[r(4.5}- r(4}], so we draw a vector in the same
0.5
direction but with twice the length of the vector r(4.5}- r(4).
r ( 4·2) - r ( 4 ) = 5[r( 4.2) - r ( 4)], so we draw a vector in the same
0.2
direction but with 5 times the (ength of the vector r( 4.2) - r( 4).
(c) By Definition I, r'(4) = J..i~
r( 4 + ~ - r ( 4 ). T (4) = ~ ~~!~I·
(d) T( 4) is a unit vector .in the same direction as r' ( 4), that is, parallel to the
tangent line to the curve at r(4) with length 1.
3. Since (x + 2) 2 = t 2 = y - 1 =>
y = (x + 2? + 1, the curve is a
parabola.
(a), (c)
(b) r'(t) = (1, 2t},
r'(- 1) = (1,·-2)
)(.
5. x = sin t, y = 2 cos t so
x 2 + (y/ 2) 2 = 1 and the curve is
an ellipse.
(a), (c)
(b) r'(t) ~ cost i - 2sintj,
r '(71")- 4 -21
v'2.
-y~
'2
J
.
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